I have some very old ones where a brass tube is inserted for the lampholder to screw on. Sometimes a nut is poked up through the base of potery/vase lamps (this is very difficult to do without the factory tools)

Any exposed metal in the stand or lampholder means it must be earthed.

Old one is just ancient. You can tell from the cable that it's out of the ark. Bought it all off Ebay and at first the plug was incorrectly wired, which tripped the fuses. DH fixed that and I just want to replace the lot tbh.

I am concerned that you might not find a lampholder which is an exact fit.

Having assumed that your lampstand is metal, wherever the flex passes through a hole in the metal, the hole must be lined with an open grommet to prevent the metal chafing through the insulation and causing your dependents to make a premature claim on your assurance policy.

the lampholders you have shown are pendant type, suitable for dangling limply from a ceiling, not suitable for a standard lamp, where it has to stand firm and erect.

I am going to guess that your standard lamp is metal, therefore it must be earthed, therefore you will need a 3-core flex and you will need a brass lampholder that attaches with some kind of nut and screw thread, and has a terminal for the earth wire to be connected. There might be a terminal on the lamp itself, near the lampholder. The screw fitting for the lampholder to the standard might be peculiar.